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## Main Question or Discussion Point

I was thinking of a Fermi-question: the thickness of atmosphere with diffusive equilibrium. And I estimated roughly

I had a lot of fun, and I am looking for interesting ways other than chemical potential to estimate the thickness of earth's atmosphere.

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Basically, I started with assumptions:

- thermal equilibrium

- diffusive equilibrium

such that I can set up the difference in chemical potential is zero, that is equal to the difference in external chemical potential (gravitational potential) plus the difference in internal potential (assuming air molecular just ideal gas); mathematically:

$$\Delta\mu=0=\Delta\mu_{ext}+\Delta\mu_{int}$$

which is

$$0=\Delta\mu_{gra}+\Delta\mu_{indeal}$$

$$0=-GM_{earth}m\Big( \frac{1}{R+\Delta h} - \frac{1}{R} \Big)+k_BT\ln\Big(\frac{n_2}{n_1}\Big)$$

Setting the ratio of

*10^{5}m*(where it should be ~*10^{4}m*). The difference of order of magnitude to real thickness is*1*(from Wiki).I had a lot of fun, and I am looking for interesting ways other than chemical potential to estimate the thickness of earth's atmosphere.

---

Basically, I started with assumptions:

- thermal equilibrium

- diffusive equilibrium

such that I can set up the difference in chemical potential is zero, that is equal to the difference in external chemical potential (gravitational potential) plus the difference in internal potential (assuming air molecular just ideal gas); mathematically:

$$\Delta\mu=0=\Delta\mu_{ext}+\Delta\mu_{int}$$

which is

$$0=\Delta\mu_{gra}+\Delta\mu_{indeal}$$

$$0=-GM_{earth}m\Big( \frac{1}{R+\Delta h} - \frac{1}{R} \Big)+k_BT\ln\Big(\frac{n_2}{n_1}\Big)$$

Setting the ratio of

*n_2, n_1*to be 0.1 (so I defined the height where density of air goes from unity - ground - to 0.1 as "thickness"), and*m*the mass of air molecular to be 10^{-26}kg,*T*to be 300K, then we will eventually obtain: $$\Delta h \approx 10^5m$$